A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216

Offset: 1

Author

Gus Wiseman, Mar 07 2019

Keywords

• nonn

Comments

These are a kind of self-describing numbers (cf. A001462, A304679).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.

Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also products of elements of A062457.

Examples

			The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  162: {1,2,2,2,2}
  250: {1,3,3,3}
  256: {1,1,1,1,1,1,1,1}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.

Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.

Range of values of A090884.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Programs

Maple

q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Mar 08 2019

Mathematica

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)

Formula

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020